How to align the material with the mesh
Unfortunately the rim of a Solidified mesh doesn't exist in time to be UV mapped. It has no UV map at all; any texture can only be projected through the solid from the faces that are UV mapped before solidifying, which gives no opportunity to change the colour on the way through.
You have some options:

Apply the Solidify modifier, and UV unwrap the whole arrow, with its (now real) rim.
Solidify the other way, so the wide sides are now the Solidify-created rim. Then you could UV map the thin sides of the rectangular profile, before solidifying.
Create a copy of the arrow after solidifying, but before any other modifier, unwrap the copy, and use a Data Transfer modifier to pick up the UV's from the copy, on the way down the original arrow's stack.

That's the option that's illustrated here (the copy is not rendered):

It would all depend on how non-destructive you need to be.. which modifiers you might need to leave yourself free to adjust.